Dehn surgery

Dehn surgery

A Dehn surgery is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. Dehn surgery can be thought of as a two stage process: drilling and Dehn filling.

In the drilling process one removes an open tubular neighbourhood of the link from the 3-manifold. The resulting manifold we call the link complement.

Given a 3-manifold with torus boundary components, we may glue in a solid torus by a homeomorphism of its boundary to the torus boundary component T of the original 3-manifold. There are many inequivalent ways of doing this, in general. This process is called Dehn filling.

Dehn surgery on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with Dehn filling on all the components of the boundary corresponding to the link.

We can pick two oriented simple closed curvesm and l on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve gamma on that torus two coordinates p and q, each coordinate corresponding to the algebraic intersection of the curve with m and l respectively. These coordinates only depend on the homotopy class of gamma.

We can specify a homeomorphism of the boundary of a solid torus to T by having the meridian curve of the solid torus map to a curve homotopic to gamma. As long as the meridian maps to the surgery slope[gamma], the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio p/q is called the surgery coefficient.

In the case of links in the 3-sphere or more generally an oriented homology sphere, there is a canonical choice of the meridians and longitudes of T given by a Seifert surface. When the ratio p/q are all integers, the surgery is called an integral surgery or a genuine surgery, since such surgeries are closely related to handlebodies and cobordism.